The star says such images put his two children in danger, particularly due to his wife's line of work.
Amal Clooney is a human rights lawyer who, the actor notes in his letter, puts terrorist groups on trial.
His comments were published by several trade magazines, including Variety, The Hollywood Reporter and Deadline.
The actor, who has starred in Gravity and Ocean's Eleven, said he and his wife are not on social media and are protective of their children's privacy.
Clooney said he decided to write the letter after seeing photos of US actress Billie Lourd's one-year-old daughter published online.
The letter is addressed at "the Daily Mail and other publications". The Daily Mail has not yet publicly responded.
The 60-year-old actor previously rejected an apology from Mail Online, the newspaper's website, after it printed a false story about Amal's mother Baria Alamuddin.
George and Amal Clooney are parents to twins who were born in 2017. The couple have been married since 2014.
"Having just seen photos of Billie Lourd's one-year-old baby in your publication, and the fact that you subsequently took those pictures down, we would request that you refrain from putting our children's faces in your publication.
"I am a public figure and accept the oftentimes intrusive photos as part of the price to pay for doing my job. Our children have made no such commitment.
"The nature of my wife's work has her confronting and putting on trial terrorist groups and we take as much precaution as we can to keep our family safe. We cannot protect our children if any publication puts their faces on their cover.
"We have never sold a picture of our kids, we are not on social media and never post pictures because to do so would put their lives in jeopardy. Not paranoid jeopardy but real world issues, with real world consequences.
"We hope that you would agree that the need to sell advertisement isn't greater than the need to keep innocent children from being targeted.
"Thank you, George Clooney."